I'm learning predicate logic, and one of the tasks is to translate English statements into predicate logic. I'm struggling with understanding how specific you have to be.
Suppose you have a statement like "there exists a person who cannot run".
Are these both valid representations of that statement:
∃x P(X) ∧ ¬R(X) where P(x) means x is a person, and R(x) means x can run.
∃x ¬R(x) where R(x) means x can run, and the domain of x is all people
Another example with a more complex statement could be "Everybody knows someone who is a runner":
∀x∃y P(x) ∧ P(y) ∧ R(y) ∧ KN(x, y) where P(x) means x is a person, R(x) means x can run, and KN(x, y) means x and y know each other
∀x∃y R(y) ∧ KN(x, y) where R(x) means x can run, and KN(x, y) means x and y know each other and the domain of x and y is all people.
Are both of these valid and representative of the statement?